The author is definitely claiming that it's not just about naming conventions: "These different perspectives ultimately amount, I argue, to mathematically inequivalent structural conceptions of the complex numbers". So you would need to argue against the substance of the article to have a basis for asserting that it is just about naming conventions.
Article: "They form the complex field, of course, with the corresponding algebraic structure, but do we think of the complex numbers necessarily also with their smooth topological structure? Is the real field necessarily distinguished as a fixed particular subfield of the complex numbers? Do we understand the complex numbers necessarily to come with their rigid coordinate structure of real and imaginary parts?"
So yes these are choices. If I care how the complex plane maps onto some real number somewhere, then I have to pick a mapping. "Real part" is only one conventional mapping. Ditto the other stuff: If I'm going to do contour integrals then I've implied some things about metric and handedness.
I still don't see how this really puts mathematicians in "disagreement." Let's pedestrian example:
I usually make an x,y plot with the x-axis pointing to the right and the y-axis pointing away from me. If I put a z-axis, personally I'll make it upwards out of the paper (sometimes this matters). Usually, but not always, my co-ordinates are meant to be smooth. But if somebody does some of this another way, are they really disagreeing with me? I think "no." If we're talking about the same problem, we'll eventually get the same answer (after we each fix 3 or 4 mistakes). If we're talking about different problems, then we need our answers to potentially "disagree."
Exactly. So I feel like all the article really says is that "Complex numbers" doesn't necessarily tell you everything you need to know. It depends on what you're doing with them.
